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Theorem abssf 39813
Description: Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
abssf.1 𝑥𝐴
Assertion
Ref Expression
abssf ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Proof of Theorem abssf
StepHypRef Expression
1 abssf.1 . . . 4 𝑥𝐴
21abid2f 2930 . . 3 {𝑥𝑥𝐴} = 𝐴
32sseq2i 3772 . 2 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ {𝑥𝜑} ⊆ 𝐴)
4 ss2ab 3812 . 2 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ ∀𝑥(𝜑𝑥𝐴))
53, 4bitr3i 266 1 ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1630  wcel 2140  {cab 2747  wnfc 2890  wss 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-in 3723  df-ss 3730
This theorem is referenced by:  rabssf  39820
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